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arXiv:2604.12796 · Physical Review A

Entanglement concentration via measurement:- role of imaginarity

Indranil Biswas, Subrata Bera, Ujjwal Sen, Indrani Chattopadhyay, Debasis Sarkar

quant-phSubmitted Apr 14, 2026• Announced Apr 15, 2026• 1 citation
arXivHTMLPDF

Abstract

The role of complex numbers in quantum theory extends beyond mathematical convenience, having recently been formalized as a resource under the framework of the resource theory of imaginarity. Operationally, imaginarity translates into using fewer resources in optical setups. In this work, we investigate the operational advantage offered by complex-valued measurements in the entanglement of assistance protocol for three-qubit systems. We demonstrate that employing such measurement bases leads to a significant improvement in the concentration of bipartite entanglement with the aid of the third party. We further analyze a modified entanglement swapping protocol and show that a three-qubit complex measurement bases with certain symmetries outperform the standard GHZ-basis. This is also one example where a three-qubit non-maximally entangled basis surpasses a maximally entangled one in generating entanglement. Construction of the basis also addresses the open problems raised in [Phys. Rev. A. \textbf{108}, 022220 (2023)]. As an intriguing application, we show that using this approach in quantum network percolation on a honeycomb lattice reduces the required bond occupation probability by $22.7\%$ and, requirement of entanglement by $10.6\%$ in each bond.

Journal reference
Phys. Rev. A 113, 062439 (2026)
DOI
10.1103/jld4-y6q5
Comments
15 pages, 4 figures, latex2e, comments welcome

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The role of complex numbers in quantum theory extends beyond mathematical convenience, having recently been formalized as a resource under the framework of the resource theory of imaginarity. Operationally, imaginarity translates into using fewer resources in optical setups. In this work, we investigate the operational advantage offered by complex-valued measurements in the entanglement of assistance protocol for three-qubit systems. We demonstrate that employing such measurement bases leads to a significant improvement in the concentration of bipartite entanglement with the aid of the third party. We further analyze a modified entanglement swapping protocol and show that a three-qubit complex measurement bases with certain symmetries outperform the standard GHZ-basis. This is also one example where a three-qubit non-maximally entangled basis surpasses a maximally entangled one in generating entanglement. Construction of the basis also addresses the open problems raised in [Phys. Rev. A. \textbf{108}, 022220 (2023)]. As an intriguing application, we show that using this approach in quantum network percolation on a honeycomb lattice reduces the required bond occupation probability by 22.7%22.7\%22.7% and, requirement of entanglement by 10.6%10.6\%10.6% in each bond.
Complex numbers are central to the formulation of quantum mechanics, yet their role as a genuine resource is only beginning to be understood. In this work, we demonstrate that quantum states with intrinsically complex amplitudes provide a fundamental advantage in state discrimination. We construct a set of five orthogonal three qubit pure states and show that the set is strongly nonlocal if and only if it includes imaginary components. Such a set becomes locally indistinguishable not only under local measurements but also against bipartite joint measurements. This enhanced robustness makes imaginarity a valuable resource for quantum cryptography since information encoded in these states remains secure against collaborative group attacks. Our results highlight a new operational role of complex numbers in quantum theory and establish imaginarity as a key enabler of cryptographic security. However, we reconstruct the set by replacing the only product state with a biseparable state whose shared entanglement between two parties nullifies the effect of imaginarity in exhibiting strong nonlocality. In fact, we show how entangling correlations between two distant parties can dilute the effect of imaginarity, and conversely, how imaginarity itself can mimic the role of entanglement. Nevertheless, the set spans a locally indistinguishable subspace, while its complement, in turn, produces distillable genuine entanglement. Notably, this is the smallest possible Unextendible Biseparable Basis (UBB) that resolves the open problem regarding the existence of a UBB of cardinality d2+d−1d^2+d-1d2+d−1 in d⊗3d^{\otimes 3}d⊗3. Our construction yields a highly powerful set, rich in resources from multiple perspectives of quantum information theory, including many-copy discrimination, unambiguous identification, entanglement creation from product state, and non-entangling perturbations.
Abstract The resource theory of imaginarity studies the operational value of imaginary parts in quantum states, operations, and measurements. Here we introduce and study the distillation and conversion of imaginarity in distributed scenario. This arises naturally in bipartite systems where both parties work together to generate the maximum possible imaginarity on one of the subsystems. We give exact solutions to this problem for general qubit states and pure states of arbitrary dimension. We present a scenario that demonstrates the operational advantage of imaginarity: the discrimination of quantum channels without the aid of an ancillary system. We then link this scenario to local operations and classical communications(LOCC) discrimination of bipartite states. We experimentally demonstrate the relevant assisted distillation protocol, and show the usefulness of imaginarity in the aforementioned two tasks.
Complex numbers are widely used in both classical and quantum physics and are indispensable components for describing quantum systems and their dynamical behavior. Recently, the resource theory of imaginarity has been introduced, allowing for a systematic study of complex numbers in quantum mechanics and quantum information theory. In this work we develop theoretical methods for the resource theory of imaginarity, motivated by recent progress within theories of entanglement and coherence. We investigate imaginarity quantification, focusing on the geometric imaginarity and the robustness of imaginarity, and apply these tools to the state conversion problem in imaginarity theory. Moreover, we analyze the complexity of real and general operations in optical experiments, focusing on the number of unfixed wave plates for their implementation. We also discuss the role of imaginarity for local state discrimination, proving that any pair of real orthogonal pure states can be discriminated via local real operations and classical communication. Our study reveals the significance of complex numbers in quantum physics and proves that imaginarity is a resource in optical experiments.
Imaginarity has shown to be an important resource in quantum information. The witness theory of quantum resource, such as entanglement witness, coherence witness, and imaginarity witness, has been established, in particular entanglement witness and coherence witness have been extensively explored. We present here another class of imaginarity witnesses beyond the one in [{Phys. Lett. A} \textbf{530}, 130135 (2025)]. Within our framework, any nonreal Hermitian operator under the reference basis is an imaginarity witness and only finite of them can detect all the imaginarity states. As the common approaches that were explored in the witness theories of entanglement and coherence, we then explore the relations between these imaginarity witnesses in different cases: (i) when they can detect common imaginary states, (ii) when they can detect the same sets of imaginary states, and (ii) when they obey the finer relation. Finally, we define an imaginarity measure in terms of witnesses, termed witnessed imaginarity, and prove that it coincides with both the trace norm of imaginarity and the robustness of imaginarity.
Wave-particle duality is one of the basic features of quantum mechanics, giving rise to the use of complex numbers in describing states of quantum systems and their dynamics and interaction. Since the inception of quantum theory, it has been debated whether complex numbers are essential or whether an alternative consistent formulation is possible using real numbers only. Here, we attack this long-standing problem theoretically and experimentally, using the powerful tools of quantum resource theories. We show that, under reasonable assumptions, quantum states are easier to create and manipulate if they only have real elements. This gives an operational meaning to the resource theory of imaginarity. We identify and answer several important questions, which include the state-conversion problem for all qubit states and all pure states of any dimension and the approximate imaginarity distillation for all quantum states. As an application, we show that imaginarity plays a crucial role in state discrimination, that is, there exist real quantum states that can be perfectly distinguished via local operations and classical communication but that cannot be distinguished with any nonzero probability if one of the parties has no access to imaginarity. We confirm this phenomenon experimentally with linear optics, discriminating different two-photon quantum states by local projective measurements. Our results prove that complex numbers are an indispensable part of quantum mechanics.
Bell-state measurements are essential ingredients in many protocols for quantum information processing, ranging from quantum teleportation and dense coding to entanglement distribution in quantum networks. Their power relies on the fact that they are measurements in an entangled basis of a two-particle system and that the used Bell-state basis can be generated from a single Bell state by local unitary transformations. How can these measurements be generalized to more particles? We develop a general framework for this state-to-measurement problem: We introduce a hierarchy of classes of measurement bases, distinguished by the local transformations the parties may use for their generation from a single state. This leads to a generalization of the concept of maximally entangleable (or weighted hypergraph) states and the identification of a novel maximally entangled basis of four qubits, being a candidate for data-hiding tasks or distillation protocols. Finally, we prove that not all forms of entanglement can be encoded in an entire measurement basis.